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Math Help - Axiom of Union and Powerset

  1. #1
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    Axiom of Union and Powerset

    Any thoughts on how to solve this one?

    Prove that for all x, UP(x) = x
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  2. #2
    Member
    Joined
    Aug 2009
    Posts
    125
    Hi,
    it is only about verifying definitions:
    y \in x implies \{y\} \in \mathcal{P}(x), and this implies y \in \cup \mathcal{P}(x) (because union of a set contains elemets of the set's elements).
    On the other hand,
    y \in \cup \mathcal{P}(x) implies \left(\exists z\right) y \in z  \&  z \in \mathcal{P}(x), and this implies y \in x (because z \in \mathcal{P}(x) means  z \subseteq x).
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